Editorial Director: Chris Hoag Editor in Chief: Deirdre Lynch Acquisitions Editor: Patrick Barbera Assistant Acquisitions Editor, Global Edition: Aditee Agarwal Editorial Assistant: Justin Billing Program Manager: Chere Bemelmans Project Manager: Christine Whitlock Program Management Team Lead: Karen Wernholm Project Management Team Lead: Peter Silvia Project Editor, Global Edition: K.K. Neelakantan Senior Manufacturing Controller, Global Edition: Trudy Kimber Media Production Manager, Global Edition: Vikram Kumar Media Producer: Aimee Thorne MathXL Content Manager: Bob Carroll Product Marketing Manager: Tiffany Bitzel Field Marketing Manager: Evan St. Cyr Marketing Coordinator: Brooke Smith Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Project Manager: Gina Cheselka Procurement Specialist: Carol Melville Associate Director of Design USHE EMSS/HSC/EDU: Andrea Nix Program Design Lead: Heather Scott Cover Image: Chill Chillz / Shutterstock.com Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com c Pearson Education Limited 2016 The rights of Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying Ye to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Authorized adaptation from the United States edition, entitled Probability & Statistics for Engineers & Scientists,9 th Edition MyStatLab Update, ISBN 978-0-13-411585-6, by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying Ye published by Pearson Education c 2017. Acknowledgements of third party content appear on page 18, which constitutes an extension of this copyright page. PEARSON, ALWAYS LEARNING and MYSTATLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the U.S. and/or other countries. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10987654321 ISBN 10: 1292161361 ISBN 13: 9781292161365 Typeset by Aptara Printed and bound in Italy by LEGO
Exercises 243 7.5 If the variable X has the probability of { 1, 0 <x<1 0 elsewhere, Show that the random variable Y = 2log e X has an exponential with λ = 1 2. 7.6 Given the random variable X with probability { e x, 0 <x< 0, elsewhere, find the probability function of Y = 3X +5. 7.7 The speed of a molecule in a uniform gas at equilibrium is a random variable V whose probability is given by { kv 2 e bv2, v > 0, f(v) = 0, elsewhere, where k is an appropriate constant and b depends on the absolute temperature and mass of the molecule. Find the probability of the kinetic energy of the molecule W,whereW = mv 2 /2. 7.8 A dealer s profit, in units of $5000, on a new automobile is given by Y = X 2,whereX is a random variable having the density function { 2(1 x), 0 <x<1, (a) Find the probability density function of the random variable Y. (b) Using the density function of Y, find the probability that the profit on the next new automobile sold by this dealership will be less than $500. 7.9 The hospital period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X +4,whereX has the density function { 32, x > 0, (x +4) 3 (a) Find the probability density function of the random variable Y. (b) Using the density function of Y, find the probability that the hospital period for a patient following this treatment will exceed 8 days. 7.10 The random variables X and Y, representing the weights of creams and toffees, respectively, in 1- kilogram boxes of chocolates containing a mixture of creams, toffees, and cordials, have the joint density function { 24xy, 0 x 1, 0 y 1,x+ y 1, f(x, y) = (a) Find the probability density function of the random variable Z = X + Y. (b) Using the density function of Z, find the probability that, in a given box, the sum of the weights of creams and toffees accounts for at least 1/2 but less than 3/4 of the total weight. 7.11 The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Assume that the joint density function of these variables is given by { 2, 0 <x<y, 0 <y<1, f(x, y) = Find the probability density function for the amount of kerosene left in the tank at the end of the day. 7.12 Let X 1 and X 2 be independent random variables each having the probability { e x, x > 0, Show that the random variables Y 1 and Y 2 are independent when Y 1 = X 1 + X 2 and Y 2 = X 1 /(X 1 + X 2 ). 7.13 A current of I amperes flowing through a resistance of R ohms varies according to the probability { 6i(1 i), 0 <i<1, f(i) = If the resistance varies independently of the current according to the probability { 2r, 0 <r<1, g(r) = 0, elsewhere, find the probability for the power W = I 2 R watts. 7.14 Let X be a random variable with probability { 1+x, 1 <x<1, 2 Find the probability of the random variable Y = X 2.
244 Chapter 7 Functions of Random Variables (Optional) 7.15 Let X have the probability { 2(x +1), 1 <x<2, 9 Find the probability of the random variable Y = X 2. 7.16 Show that the rth moment about the origin of the gamma is μ r = βr Γ(α + r). Γ(α) [Hint: Substitute y = x/β in the integral defining μ r and then use the gamma function to evaluate the integral.] 7.17 A random variable X has the discrete uniform { 1, x =1, 2,...,k, f(x; k) = k Show that the moment-generating function of X is M X (t) = et (1 e kt ) k(1 e t ). 7.18 A random variable X has the geometric g(x; p) =pq x 1 for x =1, 2, 3,... Show that the moment-generating function of X is M X (t) = pet, t < ln q, 1 qet and then use M X (t) to find the mean and variance of the geometric. 7.19 A random variable X has the Poisson p(x; μ) =e μ μ x /x! forx =0, 1, 2,... Show that the moment-generating function of X is M X (t) =e μ (e t 1). Using M X (t), find the mean and variance of the Poisson. 7.20 The moment-generating function of a certain Poisson random variable X is given by M X (t) =e 9(e t 1). Find P (μ σ X μ + σ). 7.21 Show that the moment-generating function of the random variable X having a chi-squared with v degrees of freedom is M X (t) =(1 2t) v/2. 7.22 Using the moment-generating function of Exercise 7.21, show that the mean and variance of the chisquared with v degrees of freedom are, respectively, v and 2v. 7.23 Both X and Y independently follow a geometric with a probability mass function of P (X = r) =P (Y = r) =q r p, r =0, 1, 2,...,where,p and q are positive numbers such that p + q =1.Find (a) the probability of U = X + Y ; (b) the conditional of X/(X + Y )=u. 7.24 By expanding e tx in a Maclaurin series and integrating term by term, show that M X (t) = e tx f(x) dx =1+μt + μ 2 t 2 2! + + μ r t r r! +.
Chapter 8 Fundamental Sampling Distributions and Data Descriptions 8.1 Random Sampling Populations and Samples The outcome of a statistical experiment may be recorded either as a numerical value or as a descriptive representation. When a pair of dice is tossed and the total is the outcome of interest, we record a numerical value. However, if the students of a certain school are given blood tests and the type of blood is of interest, then a descriptive representation might be more useful. A person s blood can be classified in 8 ways: AB, A, B, or O, each with a plus or minus sign, depending on the presence or absence of the Rh antigen. In this chapter, we focus on sampling from s or populations and study such important quantities as the sample mean and sample variance, which will be of vital importance in future chapters. In addition, we attempt to give the reader an introduction to the role that the sample mean and variance will play in statistical inference in later chapters. The use of modern high-speed computers allows the scientist or engineer to greatly enhance his or her use of formal statistical inference with graphical techniques. Much of the time, formal inference appears quite dry and perhaps even abstract to the practitioner or to the manager who wishes to let statistical analysis be a guide to decision-making. We begin this section by discussing the notions of populations and samples. Both are mentioned in a broad fashion in Chapter 1. However, much more needs to be presented about them here, particularly in the context of the concept of random variables. The totality of observations with which we are concerned, whether their number be finite or infinite, constitutes what we call a population. There was a time when the word population referred to observations obtained from statistical studies about people. Today, statisticians use the term to refer to observations relevant to anything of interest, whether it be groups of people, animals, or all possible outcomes from some complicated biological or engineering system. 245
246 Chapter 8 Fundamental Sampling Distributions and Data Descriptions Definition 8.1: A population consists of the totality of the observations with which we are concerned. The number of observations in the population is defined to be the size of the population. If there are 600 students in the school whom we classified according to blood type, we say that we have a population of size 600. The numbers on the cards in a deck, the heights of residents in a certain city, and the lengths of fish in a particular lake are examples of populations with finite size. In each case, the total number of observations is a finite number. The observations obtained by measuring the atmospheric pressure every day, from the past on into the future, or all measurements of the depth of a lake, from any conceivable position, are examples of populations whose sizes are infinite. Some finite populations are so large that in theory we assume them to be infinite. This is true in the case of the population of lifetimes of a certain type of storage battery being manufactured for mass throughout the country. Each observation in a population is a value of a random variable X having some probability f(x). If one is inspecting items coming off an assembly line for defects, then each observation in the population might be a value 0 or 1 of the Bernoulli random variable X with probability b(x;1,p)=p x q 1 x, x =0, 1 where 0 indicates a nondefective item and 1 indicates a defective item. Of course, it is assumed that p, the probability of any item being defective, remains constant from trial to trial. In the blood-type experiment, the random variable X represents the type of blood and is assumed to take on values from 1 to 8. Each student is given one of the values of the discrete random variable. The lives of the storage batteries are values assumed by a continuous random variable having perhaps a normal. When we refer hereafter to a binomial population, a normal population, or, in general, the population f(x), we shall mean a population whose observations are values of a random variable having a binomial, a normal, or the probability f(x). Hence, the mean and variance of a random variable or probability are also referred to as the mean and variance of the corresponding population. In the field of statistical inference, statisticians are interested in arriving at conclusions concerning a population when it is impossible or impractical to observe the entire set of observations that make up the population. For example, in attempting to determine the average length of life of a certain brand of light bulb, it would be impossible to test all such bulbs if we are to have any left to sell. Exorbitant costs can also be a prohibitive factor in studying an entire population. Therefore, we must depend on a subset of observations from the population to help us make inferences concerning that same population. This brings us to consider the notion of sampling. Definition 8.2: A sample is a subset of a population. If our inferences from the sample to the population are to be valid, we must obtain samples that are representative of the population. All too often we are
8.2 Some Important Statistics 247 tempted to choose a sample by selecting the most convenient members of the population. Such a procedure may lead to erroneous inferences concerning the population. Any sampling procedure that produces inferences that consistently overestimate or consistently underestimate some characteristic of the population is said to be biased. To eliminate any possibility of bias in the sampling procedure, it is desirable to choose a random sample in the sense that the observations are made independently and at random. In selecting a random sample of size n from a population f(x), let us define the random variable X i, i =1, 2,...,n, to represent the ith measurement or sample value that we observe. The random variables X 1,X 2,...,X n will then constitute a random sample from the population f(x) with numerical values x 1,x 2,...,x n if the measurements are obtained by repeating the experiment n independent times under essentially the same conditions. Because of the identical conditions under which the elements of the sample are selected, it is reasonable to assume that the n random variables X 1,X 2,...,X n are independent and that each has the same probability f(x). That is, the probability s of X 1,X 2,...,X n are, respectively, f(x 1 ),f(x 2 ),...,f(x n ), and their joint probability is f(x 1,x 2,...,x n )=f(x 1 )f(x 2 ) f(x n ). The concept of a random sample is described formally by the following definition. Definition 8.3: Let X 1,X 2,...,X n be n independent random variables, each having the same probability f(x). Define X 1,X 2,...,X n to be a random sample of size n from the population f(x) and write its joint probability as f(x 1,x 2,...,x n )=f(x 1 )f(x 2 ) f(x n ). If one makes a random selection of n = 8 storage batteries from a manufacturing process that has maintained the same specification throughout and records the length of life for each battery, with the first measurement x 1 being a value of X 1, the second measurement x 2 a value of X 2, and so forth, then x 1,x 2,...,x 8 are the values of the random sample X 1,X 2,...,X 8. If we assume the population of battery lives to be normal, the possible values of any X i, i =1, 2,...,8, will be precisely the same as those in the original population, and hence X i has the same identical normal as X. 8.2 Some Important Statistics Our main purpose in selecting random samples is to elicit information about the unknown population parameters. Suppose, for example, that we wish to arrive at a conclusion concerning the proportion of coffee-drinkers in the United States who prefer a certain brand of coffee. It would be impossible to question every coffeedrinking American in order to compute the value of the parameter p representing the population proportion. Instead, a large random sample is selected and the proportion ˆp of people in this sample favoring the brand of coffee in question is calculated. The value ˆp is now used to make an inference concerning the true proportion p. Now, ˆp is a function of the observed values in the random sample; since many